5 pints) or briefly, 5 eighths of a gallon; and similarly for any other fraction. 127. The rules of division are purely the formal consequences of the fundamental laws of the multiplication of numbers, III, IV, W, definition IX (461), theorem XI (463), and the corresponding laws of addition and subtraction. The rules of division, or, what is the same thing, the rules of the operation of fractions, can be deduced in the same way (472) as the rules of subtraction (438, 1–5). the meaning of the symbols a, b, c, -, +, −, ab, , (372). 128. The rule governing the dependence of signs of a fraction upon the signs of its terms is deduced from the rules of the signs of products when the factors have different signs (441, 6 and 8), thus: and at – '', we can, as above, show that They follow without regard to According to the rules used in establishing the preceding The sign written before the fraction is called the sign of the fraction. Thus, if the sign of both numerator and denominator are changed, the sign of the fraction is not changed; but if the sign of either one is changed, the sign before the fraction is changed. In case the numerator or denominator is a polynomial, we must be careful, in changing the signs, to change the sign of each of its – 1 terms (% 41, 3, 4, 5). Thus, the fraction a can be written, by changing the signs of both numerator and denominator, in the b – form a. d—c 129. It follows from 441, 6, 8, that if the terms of a fraction are the indicated products of two or more parentheses, the sign of the fraction will remain the same, if the signs of an even number of the parentheses be changed, but the sign of the fraction will be changed if the signs of an odd number of parentheses be changed. If the integer in the numerator of a fraction is less than its denominator the fraction is said to be a proper, or pure fraction, and if greater, an improper fraction. REDUCTION of FRACTIONs 130. The Reduction of Fractions to their Lowest Terms. Let the line A B be divided into seven equal parts, at D, E, F, G, II, I. | | | | | | | | | | | | | | | | | | | | | | A. D E P G. II I B Then (1) A G is ; of A B. [#126] Now let each of these parts be subdivided into 3 equal parts. Then AB contains 21 of these subdivisions and A G contains 12 of them. (2) A G is . of AB. Comparing (1) and (2) it follows that 4 12. 7 T 21 That is, the value of the fraction ; is not altered by multiplying both its terms by 3, and the value of the fraction ; is not altered by dividing both of its terms by 3. 131. The result of the previous section is a particular case of the following: Theorem I. It does not alter the value of a fraction to multiply or divide both of its terms by the same quantity. On multiplying both members of the equation by c, it becomes ac = boc = beq [Ax. 3, 481; and Law XI] Hence, it follows from (1) and (2), to reduce a fraction to lower terms, divide both numerator and denominator by any factor common to both. 132. A fraction is expressed in its lowest terms if its numerator and denominator have no common factor; and therefore any fraction can be reduced to its lowest terms by dividing both numerator and denominator by their G. C. D., because it contains all the factors common to both terms of the fraction. Since in example 3 no common factor can be determined by inspection, it is necessary to determine the G. C. D. of the numerator and the denominator by the method of division. 133. When the terms of the fraction can not be readily factored, then the G. C. D. must be found by division and the terms of the fraction divided by it. |